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arXiv:cond-mat/0210544 · Journal of Optics B Quantum and Semiclassical Optics

Microscopic many-body theory of atomic Bose gases near a Feshbach resonance

R. A. Duine, Henk T. C. Stoof

Published Apr 1, 2003• 49 citations
arXivHTMLPDF

Abstract

A Feshbach resonance in the s-wave scattering length occurs if the energy of the two atoms in the incoming open channel is close to the energy of a bound state in a coupled closed channel. Starting from the microscopic Hamiltonian that describes this situation, we derive the effective atom–molecule theory for a Bose gas near a Feshbach resonance. In order to take into account all two-body processes, we have to dress the bare couplings of the atom–molecule model with ladder diagrams. This results in a quantum field theory that exactly reproduces the scattering amplitude of the atoms and the bound-state energy of the molecules. Since these properties are incorporated at the quantum level, the theory can be applied both above and below the critical temperature of the gas. Moreover, making use of the true interatomic potentials ensures that no divergences are encountered at any stage of the calculation. We also present the mean-field theory for the Bose–Einstein condensed phase of the gas.

DOI
10.1088/1464-4266/5/2/382

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arXiv:0710.5318•Physical Review A•Apr 25, 2008•12 citations
A Feshbach resonance in the s-wave scattering length occurs if the energy of the two atoms in the incoming open channel is close to the energy of a bound state in a coupled closed channel. Starting from the microscopic Hamiltonian that describes this situation, we derive the effective atom–molecule theory for a Bose gas near a Feshbach resonance. In order to take into account all two-body processes, we have to dress the bare couplings of the atom–molecule model with ladder diagrams. This results in a quantum field theory that exactly reproduces the scattering amplitude of the atoms and the bound-state energy of the molecules. Since these properties are incorporated at the quantum level, the theory can be applied both above and below the critical temperature of the gas. Moreover, making use of the true interatomic potentials ensures that no divergences are encountered at any stage of the calculation. We also present the mean-field theory for the Bose–Einstein condensed phase of the gas.
With help of a Feshbach scattering resonance it is possible to change the effective two-body interaction of ultra-cold atoms from repulsive to attractive, from weak to strong. Regular field theory for the description of Bose-Einstein condensates is based on a single interaction parameter only, the scattering length. Such a description, however, is not valid anymore for the resonance regime. We formulated a field-theory that incorporated the microscopic energy-dependent two-body interactions, that also takes pair correlations into account. This theory allowed us to explain an experiment at JILA where Ramsey fringes were observed in a ^85Rb Bose-Einstein condensate by making use of a Feshbach resonance. The theory and experiment show that coherent molecules are created, a first step in the direction of molecular BEC. For non-resonant interactions, the self-energy corresponding to the atom-atom interactions is proportional to the scattering length. Close to resonance, however, this relationship no longer applies, and the energy will level off to a finite value on resonance. We investigate this resonance self-energy, and compare it with experimental results from strongly interacting Fermi systems.
To study the behavior of atomic Fermi gases in the vicinity of Feshbach resonance, M. Holland et al. [12] and S. J. J. M. F. Kokkelmans et al. [22] proposed resonance mean field theory in which the molecules formed by Feshbach resonance are considered in the many-body Hamiltonian. Motivated by microscopic scattering theory and resonance mean field theory, we develop a thermodynamic method to study this system. In this method we treat atomic Fermi gases near resoance as a atom-molecule mixture. We can use BCS Hamiltonian and Bose- Einstein statistics to describe the behavior of atoms and molecules respectively. Chemical equilibrium can be achieved between atoms and molecules due to hypefine interaction. We use this method to study the system behavior under different temperature and detuning energy and find that at low temperature a smooth BCS-BEC crossever can occure with decreasing detuning energy. We further calculate a measurable quantity, molecular condensation fraction, and find its behavior matches qualitatively with the experimental result [16].
We present a detailed analysis of the two-channel atom-molecule effective Hamiltonian for an ultracold two-component homogeneous Fermi gas interacting near a Feshbach resonance. We particularly focus on the two-body and many-body properties of the dressed molecules in such a gas. An exact result for the many-body T matrix of the two-channel theory is derived by both considering coupled vertex equations and functional-integral methods. Making use of this result allows us to incorporate exactly into the many-body theory the two-body physics of the Feshbach scattering by means of simple analytical formulas without any fitting parameters. New interesting many-body effects are discussed in the case of narrow resonances. We give also a description of the BEC-BCS crossover above and below TC{T}_{C}TC​. The effects of different approximations for the self-energy of the dressed molecules are discussed. The single-channel results are derived as a special limit for broad resonances. Moreover, through an analytic analysis of the BEC limit, the relation between the composite boson of the single-channel model and the dressed-molecule of the two-channel model is established.
Bose-Einstein condensation (BEC) of Feshbach molecules in a homogeneous Bose gas is studied at finite temperatures in a single-channel mean-field approach where the Hartree-Fock energy and pairing gap are determined self-consistently. In the molecular-BEC state, the atomic excitation is gapped and the molecular excitation is gapless. The binding energy of Feshbach molecules is shifted from the vacuum value by many-body effects. When the scattering length as{a}_{s}as​ of the atoms is negative, the system is subject to mechanical collapse due to negative compressibility. The system is stable in most regions with positive scattering lengths. However, at low temperatures near the resonance, the molecular-BEC state vanishes, and the coherent mixture of atomic and molecular BEC is subject to mechanical collapse.
The use of Feshbach resonances in the field of ultracold gases has received a lot of attention recently. In first instance it was realized that these resonances can be used to alter the interaction strength in an atomic gas, by means of tuning an external magnetic field. Furthermore, by sweeping the magnetic field from one side of the resonance to the other, a degenerate gas of bosonic molecules can be created, a significant accomplishment. By making the interactions between fermionic atoms strongly attractive, a fermionic superfluid was recently made, a feat that has drawn a lot of attention in the physics community and from the media. We present work that describes these many-particle systems near a Feshbach resonance in a comprehensive way by using the concept of dressed molecules. These dressed molecules are essentially two-particle states that are a superposition of a tightly bound bare molecule, and two unbound bare atoms. This description reduces to the correct paired wave-function in the two-body limit. Moreover it gives a physically correct description of all the phenomena mentioned above: the Bose-condensation of a gas near a Feshbach resonance, that contains molecules, is more correctly interpreted as a Bose-condensation of dressed molecules. The fermionic superfluid contains a condensate of bosonic pairs, normally called Cooper pairs, that can be understood as the dressed molecules. For the latter case, the dressed-molecule picture also gives an elegant characterization of the crossover between a Bose condensate of pure molecules on one side of the resonance, to a fermionic Bardeen-Cooper-Schrieffer superfluid on the other side. Then, the dressed molecules change continuously from being mostly a bare molecule, to being mostly a loose pair of two atoms. It is clear that the nature of the dressed molecules is much determined by the probability for it to be in the bare-molecular state. This probability we denote with a single parameter Z. We present various ways of calculating it, and study how it affects the macroscopic physics.
In this paper, we study the phase diagram of a homogeneous Bose gas with a repulsive interaction near a wide Feshbach resonance at zero temperature. The Bose-Einstein-condensation (BEC) state of atoms is a metastable state. When the scattering length aaa exceeds a critical value depending on the atom density nnn, na3>0.035n{a}^{3}>0.035na3>0.035, the molecular excitation energy is imaginary and the atomic BEC state is dynamically unstable against molecule formation. The BEC state of diatomic molecules has lower energy, where the atomic excitation is gapped and the molecular excitation is gapless. However, when the scattering length is above another critical value, na3>0.0164n{a}^{3}>0.0164na3>0.0164, the molecular BEC state becomes an unstable coherent mixture of atoms and molecules. In both BEC states, the binding energy of diatomic molecules is reduced due to the many-body effect.